## Determination of expected portfolio return and variance

As it was established, the expected yield of a portfolio consisting of securities * n * is calculated using the formula

where * W * i is the weight of the security in the portfolio.

We substitute in this formula the expression from the formula (3.17):

Select in this equation the terms that are not affected by market changes and which depend on market indicators:

(3.21)

To make this formula compact, Sharpe proposed to consider the market index as a characteristic of the conditional (* n + * 1) -th stock in the portfolio. In this case, the second term of equation (3.21) can be represented in the form

where (3.22)

(3.23)

It is assumed that the variance of the error (* n + * 1) of the portfolio's stock is equal to the variance of the market return:

Expression (3.22) is the sum of the weighted values of the " beta " () of each security (where the weight is ) and is called * portfolio beta * (). Taking into account the expressions (3.22) and (3.23), the formula (3.21) can be written as:

(3.24)

So, the expected return on the portfolio can be represented as consisting of two parts:

a) the sum of the weighted parameters a; each foam paper - (), which reflects the contribution to the securities themselves

b) the components , i.e. products portfolio portfolio and the expected market return, which reflects the relationship of the market with securities portfolio.

Portfolio variance. As you know, the portfolio variance can be represented in the form

If instead of the values and substitute the expressions (3.19) and (3.20) into this equation:

hold the appropriate calculations and use the convention (3.22), then it can be shown that the variance of the portfolio is represented in the form

(3.25)

It must be borne in mind that i.e.

Hence, the variance of the portfolio containing * n * shares can be represented as consisting of two components:

a) the weighted average variances of errors, where the weights are , which reflects the share of the portfolio risk associated with the risk of the securities themselves (own risk);

b) - the weighted variance of the return on the market portfolio , where the square of the portfolio beta serves as the weight, determined by the instability of the market itself (market risk).

Based on the above, we can, similarly to the way it was done above, to show that with the increase in the number of securities in the portfolio, the first part of the risk of the portfolio will tend to zero. Therefore, diversification of the portfolio leads to a reduction in the risk associated with the instability of the securities themselves, leaving only the component, which depends on the instability of the market itself.

## Formulation of the investor's goal in the Sharpe model

Recall that in the Markowitz model, the investor's goal was formulated as follows.

It is necessary to find the minimum value of portfolio variance

for given initial conditions

In the Sharpe model, the investor's goal is as follows. It is necessary to find the minimum value of portfolio variance

(3.26)

under the following initial conditions:

(3.27)

(3.28)

(3.29)

In both models, the analyst aims to minimize the variance of the portfolio for a given portfolio return * E * * (the first condition), keeping in mind that the sum of the shares of initial investment, directed by him for the purchase of securities in the portfolio, is equal to one (the second condition). However, the models also have a significant difference:

• Firstly, formulas for calculating and ;

are presented in different ways

• Secondly, there is an additional fourth constraint that introduces portfolio beta as the weight of the market indicator.

## Constructing the Border of Effective Portfolios

Note the main steps that need to be taken to construct the boundaries of effective portfolios in the Sharp model.

1. Select the * n * securities from which the portfolio is formed, and determine the historical gap in the * N * calculation steps for which the yield values will be each security.

2. At the market index (for example, the MICEX index), calculate the market yields for the same period of time.

3. Determine the variance of the market portfolio , as well as the covariance of the yields of each security with market yield and find the values :

4. Find the expected returns of each security and the yield of the market portfolio and calculate the parameter :

5. Calculate variances errors of the regression model.

6. Substitute these values into equations (3.26) - (3.29).

After such a substitution, it becomes clear that the unknown values are the weights of the portfolio. Having chosen a certain value of the expected yield of the portfolio * E *, *, we can solve the system of equations (3.26) - (3.29) using the Lagrange multipliers.

Let's consider an example of constructing the boundaries of effective portfolios consisting of shares * A, B * and * C. * The investor's task in this case is as follows: it is necessary to minimize the expression

under the following initial conditions:

Substitute the previously calculated values of and into these expressions:

Solving this problem using Lagrange multipliers gives the following results:

Thus, the task was solved: for any chosen level of expected return of the portfolio * Е * * the investor can find the weights of each security and form a portfolio that has minimal risk. Hence, the investor is able to construct the border of effective portfolios, and then, having put on it a curve of indifference curves, determine the optimal portfolio.

To find the weights of Wi securities, you must first compile the Lagrange polynomial:

where - Lagrange multipliers. Then the seven partial derivatives of the polynomial * L * are taken for each of the seven unknowns , and are equated to zero :

In matrix form, these seven equations are written in the following form:

We represent this in the form of a matrix equation:

To find the weight you need to compute on the computer the matrix , the inverse of the matrix * T, * and solve equation , i.e. Each row of the inverse matrix is multiplied by the column * E. * The weights for the * MVP * of the portfolio are calculated by finding the inverse matrix , "images/image445.jpg"> is the matrix without the fifth row and the fifth column corresponding to the constraint * E *. *

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